An Additional Case: Target System Reliability is 4.0

The previous results are associated with the case in which the target system reliability index is 3.5. However, if the analyzed bridge is classified as a critical or essential bridge (i.e., it is part of a very important highway system), its designed reliability index is expected to be higher. Therefore, an additional case in which the target system reliability index is 4.0 is investigated. The redundancy factor herein is defined as the ratio of the mean resistance of a component in a system when the system reliability index is 4.0 to the mean resistance of the same component when its reliability index is 3.5.

63

By performing the same procedure as in the previous E_sys,target 3.5^{case, the} redundancy factors associated with three different systems are obtained using the idealized systems, as shown in Table 2.20. Correspondingly, the designed mean resistances of girders and the associated component and system reliability indices are calculated (see Table 2.21 and Table 2.22, respectively). It is found from these tables that (a) the redundancy factors in this case are all greater than those in the previous

sys,target 3.5

E case; this is because the target system reliability index herein is higher;

(b) all the redundancy factors are greater than 1.0; therefore, all the component reliability indices are larger than 3.5 (even in the parallel system when ρ(Ri,Rj)=0); this is different from the finding in the previous E_sys,target 3.5 case; (c) the final mean resistance of girders is still governed by the interior girder; hence, the reliability indices of exterior girders are larger than those of interior girders; and (d) the system reliability indices of all the systems meet the predefined system reliability level

sys,target 4.0

E ^.

Similar to the previous E_sys,target 3.5 case, the redundancy factors associated with the 4-component series-parallel system that considers the perfect correlation between components 2, 3 and 4, 5 are also calculated: 1.058 (no correlation case), 1.070 (partial correlation case), and 1.081 (perfect correlation case). The associated designed mean resistances of girders and the reliability indices of girders and the system for this

sys,target 4.0

E case are listed in Table 2.23 and Table 2.24, respectively. Comparing the results in Table 2.21 to Table 2.24, it is observed that the results with / without considering the perfect correlation among some components are very close and the

64

designed mean resistances of girders based on the 6-component system (without considering the perfect correlation) are slightly conservative.

2.9 CONCLUSIONS

In this chapter, a redundancy factor is proposed to provide a rational system reliability-based design of structural components. By using idealized systems consisting of identical components, the effects of the system type, correlations among the resistances of components, number of components in a system, coefficients of variation of load and resistances, and mean value of the load on the redundancy factor are investigated. For a representative case, the redundancy factors of N-component systems without considering the post-failure material behavior are evaluated with respect to different correlation cases and system types.

Next, systems consisting of two to four components are used to demonstrate the procedure for evaluating the redundancy factors of ductile, brittle, and mixed systems.

The effects of number of brittle components in a system and post-failure behavior factor on the redundancy factor are also studied using these systems. Then, the redundancy factors of N-component ductile and brittle systems with large number of components are calculated for the aforementioned representative case. Two types of limit states in which system redundancy is taken into account from the load and resistance side, respectively, are provided. Finally, a bridge example is presented to demonstrate the application of the redundancy factor. The following conclusions are drawn:

65

1. The redundancy factor ηR proposed in this chapter and the factor relating to redundancy in the AASHTO bridge design specifications are of the same nature.

The major difference is that the factor relating to redundancy in the AASHTO specifications is determined based on a general classification of redundancy levels while the proposed redundancy factor ηR in this chapter is more rational since it is based on a comprehensive system reliability-based approach considering several parameters including the system type, correlation among the resistances of components, number of components in the system, and post-failure material behavior of components.

2. An approach for simplifying the system model in the redundancy factor analysis of brittle systems is proposed. By reducing the N!×N series-parallel system model to the N×N series-parallel system model, this approach makes it possible to calculate the redundancy factor of brittle parallel systems with large number of components.

3. For the systems without considering the post-failure material behavior, (a) increasing the coefficient of variation of resistance leads to higher redundancy factors in series systems but lower redundancy factors in parallel systems; (b) as the coefficient of variation of load increases, the redundancy factors associated with both series and parallel systems increase; (c) the mean value of load has no effect on the redundancy factors; and (d) the effect of N on the redundancy factors in mp×ns series-parallel systems having the same number of parallel components (i.e., m is same in these systems) is similar to that in the series system; while the effect of N on the redundancy factors in ms×np series-parallel systems having the

66

same number of series components (i.e., m is same) is similar to that in the parallel system.

4. In the ductile case, (a) the difference in the redundancy factors between the normal and lognormal distributions is significant in the parallel system; and (b) when the number of components in the parallel system is small, increasing N leads to a significant decrease of the redundancy factor; however, as N continues increasing this decrease becomes insignificant.

5. In the brittle case, the redundancy factors associated with series, parallel, and series-parallel systems are almost the same; this indicates that for an N-component brittle structure, the redundancy factor is independent of the system type.

6. In the mixed case, the redundancy factors are at least 1.0 due to the existence of brittle component(s) in the systems. As the number of brittle components increases in an N-component mixed system, the redundancy factor becomes larger and closer to the redundancy factor associated with the brittle case. Increasing the correlation among the resistances of components leads to a lower redundancy factor in the mixed parallel systems.

7. This chapter is for codification purpose. The proposed approach can be used to calculate the redundancy factors for a wide range of systems with different number of components, different system types, and different correlation cases.

8. This chapter presents standard tables of redundancy factors associated with a representative V(R) and V(P) case. Further effort is necessary to generate standard tables with respect to different combinations of V(R) and V(P). When this

67

information becomes available, the redundancy factors corresponding to a specific system will be determined from these tables and then directly used in the design.

68

Table 2.1 Ecs(R), ηR and βcs of three-component systems when R and P follow normal distribution.

System type Correlation

Series system Parallel system Ecs(R); K_R^;Ecs ^Ecs(R); KR^;Ecs

ρ(Ri,Rj) = 0 17.685; 1.049; 3.78 13.684; 0.812; 2.17 ρ(Ri,Rj) = 0.5 17.651; 1.047; 3.77 14.817; 0.879; 2.69 ρ(Ri,Rj) = 1 16.861; 1.000; 3.50 16.861; 1.000; 3.50 Note: E(P)= 10; V(P)= 0.1; V (R)= 0.1; βc = 3.5; βsys = 3.5; Ec,N (R) = 16.861.

69

Table 2.2 Ecs(R), ηR and βcs of three-component systems when R and P follow lognormal distribution.

System type Correlation

Series system Parallel system Ecs(R); K_R^; Ecs ^Ecs(R); KR^; Ecs

ρ(Ri,Rj) = 0 17.045; 1.040; 3.78 14.092; 0.860; 2.43 ρ(Ri,Rj) = 0.5 16.985; 1.037; 3.76 14.969; 0.914; 2.86 ρ(Ri,Rj) = 1 16.384; 1.000; 3.50 16.384; 1.000; 3.50 Note: E(P)= 10; V(P)= 0.1; V (R)= 0.1; βc = 3.5; βsys = 3.5; Ec,LN (R) = 16.384.

70

Table 2.3 Ecs(R) and ηR of N-component systems using RELSYS when R and P follow normal distribution.

System Ecs(R) K_R

100-component system

Series system 24.185 1.144

5p×20s SP system 20.655 0.977

10p×10s SP system 19.618 0.928

20p×5s SP system 18.853 0.892

200-component system

Series system 24.723 1.17

5p×40s SP system 21.019 0.995

10p×20s SP system 19.915 0.942

20p×10s SP system 19.069 0.902

Note: E(P)= 10; V(P)= 0.3; V (R)= 0.05; ρ(Ri, Rj) = 0; βc = 3.5; βsys = 3.5; Ec,N (R) = 21.132.

71

Table 2.4 Ecs(R) and ηR of different systems associated with the case ρ(Ri,Rj) = 0 using the MCS-based program.

System Normal distribution Lognormal distribution

Ecs(R) K_R ^Ecs(R) K_R

72

Table 2.5 Ecs(R) and ηR of different systems associated with the case ρ(Ri,Rj) = 0.5 using the MCS-based program.

System Normal distribution Lognormal distribution Ecs(R)

K

_R ^Ecs(R)

K

R

73

Table 2.6 Ecs(R), ηR, and βcs of three- and four-component ductile systems associated with normal distribution.

Correlation

Three-component parallel system

Four-component parallel system

Four-component 2p×2s series-parallel system Ecs(R);

K

_R^;

E

cs Ecs(R);

K

_R^;

E

cs Ecs(R);

K

_R^;

E

cs

ρ(Ri,Rj) = 0 20.699; 0.980; 3.37 20.660; 0.978; 3.36 21.160; 1.001; 3.51 ρ(Ri,Rj) = 0.5 20.910; 0.989; 3.44 20.893; 0.989; 3.43 21.231; 1.005; 3.53 ρ(Ri,Rj) = 1 21.132; 1.000; 3.50 21.132; 1.000; 3.50 21.132; 1.000; 3.50 Note: V(R)= 0.05; V (P)= 0.3; βc = 3.5; βsys = 3.5; Ec (R) = 21.132

74

Table 2.7 Ecs(R), ηR, and βcs of mixed systems associated with the case ρ(Ri,Rj) = 0 when R and P are normal distributed.

System Ecs (R) K_R ^βcs

2-component

parallel system 1 ductile & 1 brittle 21.280 1.007 3.55 3-component

parallel system

1 ductile & 2 brittle 21.630 1.024 3.65 2 ductile & 1 brittle 21.300 1.008 3.55

4-component parallel system

1 ductile & 3 brittle 21.850 1.034 3.71 2 ductile & 2 brittle 21.640 1.024 3.65 3 ductile & 1 brittle 21.319 1.009 3.56

4-component series-parallel system

(2p×2s SP system)

1 ductile & 3 brittle 21.850 1.034 3.71 2 ductile & 2 brittle

Case A 21.680 1.026 3.66

2 ductile & 2 brittle

Case B 21.680 1.026 3.66

3 ductile & 1 brittle 21.440 1.015 3.59 Note: V(P)= 0.3; V (R)= 0.05; βc = 3.5; βsys = 3.5; Ec(R) = 21.132.

75

Table 2.8 Ecs(R), ηR, and βcs of mixed systems associated with the case ρ(Ri,Rj) = 0.5 when R and P are normal distributed.

System Ecs (R) K_R ^βcs

2-component

parallel system 1 ductile & 1 brittle 21.260 1.006 3.53 3-component

parallel system

1 ductile & 2 brittle 21.530 1.019 3.62 2 ductile & 1 brittle 21.290 1.007 3.55

4-component parallel system

1 ductile & 3 brittle 21.700 1.027 3.67 2 ductile & 2 brittle 21.550 1.020 3.62 3 ductile & 1 brittle 21.318 1.009 3.55

4-component series-parallel system

(2p×2s SP system)

1 ductile & 3 brittle 21.700 1.027 3.67 2 ductile & 2 brittle

Case A 21.585 1.021 3.63

2 ductile & 2 brittle

Case B 21.585 1.021 3.63

3 ductile & 1 brittle 21.420 1.014 3.59 Note: V(P)= 0.3; V (R)= 0.05; βc = 3.5; βsys = 3.5; Ec (R) = 21.132.

76

Table 2.9 Ecs(R) and ηR of ductile systems associated with the case ρ(Ri,Rj) = 0.

System Normal distribution Lognormal distribution Ecs(R) ηR Ecs(R) ηR

100-component

system

Series system 23.626 1.118 30.457 1.120 Parallel system 20.519 0.971 26.759 0.984 5p×20s SP system 21.428 1.014 27.928 1.027 10p×10s SP system 21.026 0.995 27.439 1.009 20p×5s SP system 20.794 0.984 27.085 0.996

300-component

system

Series system 24.112 1.141 31.028 1.141 Parallel system 20.498 0.970 26.759 0.984 5p×60s SP system 21.639 1.024 28.227 1.038 10p×30s SP system 21.195 1.003 27.656 1.017 20p×15s SP system 20.921 0.990 27.303 1.004

500-component

system

Series system 24.323 1.151 31.246 1.149 Parallel system 20.498 0.970 26.759 0.984 5p×100s SP system 21.745 1.029 28.309 1.041 10p×50s SP system 21.280 1.007 27.738 1.020 20p×25s SP system 20.963 0.992 27.357 1.006 Note: E(P)= 10; V(P)= 0.3; V (R)= 0.05; βc = 3.5; βsys = 3.5; Ec,N (R) = 21.132;

Ec,LN (R) = 27.194.

77

Table 2.10 Ecs(R) and ηR of ductile systems associated with the case ρ(Ri,Rj) = 0.5.

System Normal distribution Lognormal distribution Ecs(R) ηR Ecs(R) ηR

100-component

system

Series system 23.013 1.089 29.533 1.086 Parallel system 20.815 0.985 26.976 0.992 5p×20s SP system 21.449 1.015 27.847 1.024 10p×10s SP system 21.195 1.003 27.439 1.009 20p×5s SP system 21.026 0.995 27.221 1.001

300-component

system

Series system 23.309 1.103 29.913 1.100 Parallel system 20.815 0.985 26.949 0.991 5p×60s SP system 21.660 1.025 28.010 1.030 10p×30s SP system 21.322 1.009 27.602 1.015 20p×15s SP system 21.132 1.000 27.330 1.005

500-component

system

Series system 23.457 1.110 30.077 1.106 Parallel system 20.815 0.985 26.949 0.991 5p×100s SP system 21.703 1.027 28.064 1.032 10p×50s SP system 21.364 1.011 27.656 1.017 20p×25s SP system 21.132 1.000 27.357 1.006 Note: E(P)= 10; V(P)= 0.3; V (R)= 0.05; βc = 3.5; βsys = 3.5; Ec,N (R) = 21.132;

Ec,LN (R) = 27.194.

78

Table 2.11 Component reliability index βcs of ductile systems.

System Normal distribution Lognormal distribution ρ= 0 ρ= 0.5 ρ = 0 ρ = 0.5

100-component system

Series system 4.23 4.05 3.88 3.77

Parallel system 3.32 3.40 3.44 3.48

300-component

system

Series system 4.36 4.14 3.94 3.81

Parallel system 3.31 3.40 3.44 3.47

500-component

system

Series system 4.42 4.18 3.96 3.83

Parallel system 3.31 3.40 3.44 3.47

Note: ρ denotes ρ(Ri,Rj); E(P)= 10; V(P)= 0.3; V (R)= 0.05; βc = 3.5; βsys = 3.5;

Ec,N (R) = 21.132; Ec,LN (R) = 27.194.

79

Table 2.12 Ecs(R) and ηR of brittle systems associated with the case ρ(Ri,Rj) = 0.

System Normal distribution Lognormal distribution Ecs(R) ηR Ecs(R) ηR

80

Table 2.13 Ecs(R) and ηR of brittle systems associated with the case ρ(Ri,Rj) = 0.5.

System Normal distribution Lognormal distribution Ecs(R) ηR Ecs(R) ηR

81

Table 2.14 Component reliability index βcs of brittle systems.

System Normal distribution Lognormal distribution ρ = 0 ρ = 0.5 ρ = 0 ρ = 0.5 5-component

system

Series system 3.79 3.73 3.67 3.62

Parallel system 3.79 3.73 3.67 3.62

10-component system

Series system 3.90 3.81 3.72 3.67

Parallel system 3.90 3.81 3.72 3.67

15-component system

Series system 3.97 3.86 3.76 3.69

Parallel system 3.97 3.86 3.76 3.69

20-component system

Series system 4.01 3.89 3.78 3.70

Parallel system 4.01 3.89 3.78 3.70

25-component system

Series system 4.04 3.91 3.79 3.71

Parallel system 4.04 3.91 3.79 3.71

50-component system

Series system 4.14 3.98 3.84 3.74

Parallel system 4.14 3.98 3.84 3.74

Note: E(P)= 10; V(P)= 0.3; V (R)= 0.05; βc = 3.5; βsys = 3.5; Ec,N (R) = 21.132;

Ec,LN (R) = 27.194.

82

Table 2.15 The redundancy factors of the three systems.

Correlation case Series system Parallel system Series-parallel system ( ,R R_{i j}) 0

U ^{1.041 0.934 0.987}

( ,R R_{i j}) 0.5

U ^{1.032 0.956 0.995}

( ,R R_{i j}) 1.0

U ^{1.000 1.000 1.000}

Note: V(R) = 0.05; V(P) = 0.3.

83

Table 2.16 The designed mean resistances of exterior Ecs(MU,ext)and interior girders Ecs(MU,int) in the four-component systems.

System type Correlation case Ecs(MU,ext),

Note: E(ML,ext)=3407 kN∙m; E(ML,int)=3509 kN∙m; V(R)=0.05; V(P)=0.3;

Ec,N(MU,ext)=7200 kN∙m; Ec,N (MU,int)=7415 kN∙m.

84

Table 2.17 The reliability indices of exterior and interior girders and the system reliability indices.

System type Correlation case E_ext Eint Esys

Series system

( ,R R_{i j}) 0

U ^{3.95 3.75 3.58}

( ,R R_{i j}) 0.5

U 3.89 3.70 3.60

( ,R R_{i j}) 1.0

U 3.69 3.50 3.50

Parallel system

( ,R R_{i j}) 0

U 3.26 3.08 3.61

( ,R R_{i j}) 0.5

U 3.40 3.22 3.63

( ,R R_{i j}) 1.0

U 3.69 3.50 3.69

Series-parallel system

( ,R R_{i j}) 0

U 3.60 3.42 3.62

( ,R R_{i j}) 0.5

U 3.65 3.47 3.61

( ,R R_{i j}) 1.0

U 3.69 3.50 3.50

Note: E(ML,ext)=3407 kN∙m; E(ML,int)=3509 kN∙m; V(R)=0.05; V(P)=0.3;

Ec,N (MU,ext)=7200 kN∙m; Ec,N (MU,int)=7415 kN∙m.

85

Table 2.18 The designed mean resistance associated with the 4-component series-parallel system.

System type Correlation case Ecs(MU,ext), kN∙m

Ecs(MU,int), kN∙m

Ecs(MU),

kN∙m Series-parallel

system

( ,R R_{i j}) 0

U ^{7078 7289 7289}

( ,R R_{i j}) 0.5

U 7135 7348 7348

( ,R R_{i j}) 1.0

U 7200 7415 7415

Note: E(ML,ext)=3407 kN∙m; E(ML,int)=3509 kN∙m; V(R)=0.05; V(P)=0.3;

Ec,N (MU,ext)=7200 kN∙m; Ec,N (MU,int)=7415 kN∙m.

86

Table 2.19 The reliability indices of exterior and interior girders and the system reliability indices associated with the 4-component series-parallel system.

System type Correlation case E_ext Eint Esys

Series-parallel system

( ,R R_{i j}) 0

U ^{3.58 3.39 3.59}

( ,R R_{i j}) 0.5

U 3.63 3.44 3.58

( ,R R_{i j}) 1.0

U 3.69 3.50 3.50

Note: E(ML,ext)=3407 kN∙m; E(ML,int)=3509 kN∙m; V(R)=0.05; V(P)=0.3;

Ec,N (MU,ext)=7200 kN∙m; Ec,N (MU,int)=7415 kN∙m.

87

Table 2.20 The redundancy factors of the four-component systems when βsys,target = 4.0.

Correlation case Series system Parallel system Series-parallel system ( ,R R_{i j}) 0

U ^{1.123 1.004 1.062}

( ,R R_{i j}) 0.5

U ^{1.113 1.030 1.072}

( ,R R_{i j}) 1.0

U ^{1.081 1.081 1.081}

Note: V(R)=0.05; V(P)=0.3.

88

Table 2.21 The designed mean resistances of exterior Ecs(MU,ext) and interior girders Ecs(MU,int) when βsys,target = 4.0.

Note: E(ML,ext)=3407 kN∙m; E(ML,int)=3509 kN∙m; V(R)=0.05; V(P)=0.3;

Ec,N (MU,ext)=7200 kN∙m; Ec,N (MU,int)=7415 kN∙m.

89

Table 2.22 The reliability indices of exterior and interior girders and the system reliability indices when βsys,target = 4.0.

System type Correlation case E_ext Eint Esys

Series system

( ,R R_{i j}) 0

U ^{4.46 4.26 4.08}

( ,R R_{i j}) 0.5

U 4.40 4.20 4.08

( ,R R_{i j}) 1.0

U 4.20 4.00 4.00

Parallel system

( ,R R_{i j}) 0

U 3.71 3.53 4.11

( ,R R_{i j}) 0.5

U 3.88 3.69 4.12

( ,R R_{i j}) 1.0

U 4.20 4.00 4.20

Series-parallel system

( ,R R_{i j}) 0

U 4.08 3.88 4.11

( ,R R_{i j}) 0.5

U 4.14 3.95 4.10

( ,R R_{i j}) 1.0

U 4.20 4.00 4.00

Note: E(ML,ext)=3407 kN∙m; E(ML,int)=3509 kN∙m; V(R)=0.05; V(P)=0.3;

Ec,N (MU,ext)=7200 kN∙m; Ec,N (MU,int)=7415 kN∙m.

90

Table 2.23 The designed mean resistance associated with the 4-component series-parallel system when βsys,target = 4.0.

System type Correlation case Ecs(MU,ext), kN∙m

Ecs(MU,int), kN∙m

Ecs(MU),

kN∙m Series-parallel

system

( ,R R_{i j}) 0

U ^{7618 7845 7845}

( ,R R_{i j}) 0.5

U 7701 7931 7931

( ,R R_{i j}) 1.0

U 7782 8014 8014

Note: E(ML,ext)=3407 kN∙m; E(ML,int)=3509 kN∙m; V(R)=0.05; V(P)=0.3;

Ec,N (MU,ext)=7200 kN∙m; Ec,N (MU,int)=7415 kN∙m.

91

Table 2.24 The reliability indices of exterior and interior girders and the system reliability indices associated with the 4-component series-parallel system when βsys,target = 4.0.

System type Correlation case E_ext Eint Esys

Series-parallel system

( ,R R_{i j}) 0

U 4.05 3.86 4.08

( ,R R_{i j}) 0.5

U 4.13 4.24 4.09

( ,R R_{i j}) 1.0

U 4.20 4.00 4.00

Note: E(ML,ext)=3407 kN∙m; E(ML,int)=3509 kN∙m; V(R)=0.05; V(P)=0.3;

Ec,N (MU,ext)=7200 kN∙m; Ec,N (MU,int)=7415 kN∙m.

92

Figure 2.1 Flowchart of the procedure for determining the redundancy factor ηR.

Given: E(P), V(P), V(R) To achieve: β_c=3.5

Obtain the mean resistance of the component, E_c(R)

Given: Distribution type of R and P, E(P), V(P), V(R), ρ(R_i, R_j)

To achieve: β_sys=3.5

Obtain the mean component resistance in the system, E_cs(R)

Using RELSYS

or MCS-based program

Redundancy factor η_R= E_cs(R) / E_c(R)

For a single component For a system with N identical components

93

Figure 2.2 Three-component systems: (a) series system; and (b) parallel system.

component 2

component 1 component 3

component 1

component 2

component 3 (a)

(b)

94

Coefficient of variation of resistance, V(R) Two-component system

Coefficient of variation of load, V(P) No correlation Perfect correlation Parallel system

Series system

95

Coefficient of variation of resistance, V(R) Two-component system

Coefficient of variation of load, V(P) (b)

96

Coefficient of variation of resistance, V(R) Three-component system

Coefficient of variation of load, V(P) Series system

97

Figure 2.6 Four-component systems: (a) series system; (b) parallel system; and (c) series-parallel system.

(a)

(b) (c)

component 2

component 1 component 3 component 4

component 1

component 2

component 3

component 4

component 1

component 2

component 3

component 4

98

Figure 2.7 Effects of V(R) on ηR in four-component systems associated with the case of (a) no correlation; (b) partial correlation; and (c) perfect correlation.

0.70

Coefficient of variation of resistance, V(R) Parallel system

Coefficient of variation of resistance,V(R) Four-component system

Coefficient of variation of resistance,V(R) Four-component system

E(P)=10; V(P)=0.1;

ρ(Ri,Rj)=1.0;βc= βsys=3.5

All systems (c)

99

Figure 2.8 Effects of V(P) on ηR in four-component systems associated with the case of (a) no correlation; (b) partial correlation; and (c) perfect correlation.

0.90

Coefficient of variation of load, V(P) 0.70

Coefficient of variation of load, V(P) Series system

Coefficient of variation of load, V(P) Four-component system

100

Figure 2.9 Effects of number of components on ηR with the variations of (a) V(R); (b) V(P); and (c) E(P) in two extreme correlation cases.

0.70

Coefficient of variation of resistance, V(R) Series systems

Coefficient of variation of load, V(P) Parallel systems

Mean value of load, E(P) Series systems

Two components Three components Four components One component

One component

Two components Three components Four components

101

Figure 2.10 Flowchart for the algorithm combined with RELSYS.

Starting from x₁, find the x_ithat satisfies:

( β_sys| x_i ) < 3.5 and ( β_sys| x_i+1) > 3.5 Given: E(P), V(R), V(P), ρ(R_i,R_j), N, distribution type of R and P, system type,

and a group of initial guess for E_cs(R):

x=[x₁, x₂, x₃, …, x_k];

also define k = 20; c = 0; Tol = 10^-4

E_cs(R) = x_i or xi+1

c = c+1;

clear the original x vector Is |( β_sys| x_i ) - 3.5| < Tol

or

|( β_sys| x_i+1) - 3.5| < Tol

?

Use x_i as the first element to generate a new x vector:

x=[x_i, x_i +10^-c, …, x_i + (k-1)×10^-c];

YES

NO

102

Figure 2.11 Schematic figure of (a) mp×ns series-parallel system (n series of m components in parallel); and (b) ms×np series-parallel system (n parallel of m components in series).

component 1

component 2

component m

…

component (m+1)

component (m+2)

component 2m

…

component [(n-1)m+1]

component [(n-1)m+2]

component n×m

… …

component 1 component 2 component m

…

… … …

component (m+1) component 2m

component (m+2)

…

component [(n-1)m+1] component n×m

component [(n-1)m+2]

…

(a)

(b)

103

Figure 2.12 Flowchart for the algorithm combined with MCS-based program.

Generate w random samples for each Riand P Given: E(P), V(R), V(P), ρ(Ri,Rj), N, distribution type of R and P, system type, number of simulation samples w, and an initial guess for

Ecs(R); define c = 1

Return the initial value as final E_cs(R) Obtain the limit state equations of each

component: g_i= R_i- P Find the average failure probability and convert it to reliability index β_sys Series Adjust the initial value for E_cs(R);

define c = 1

104

Figure 2.13 The effects of number of component on (a) component reliability index βcs; and (b) redundancy factor ηR (Note: “N” is normal distribution; “LN”

Number of components, N LN-0.5

Number of components, N

N-0.5 LN-0 LN-0.5

N-0

All systems, perfect correlation LN-0.5 N-0.5 N-0

105

Figure 2.14 Failure modes of three-component brittle parallel system.

comp 1: g₁

comp 2|1: g₄

comp 3|1,2: g₁₀

comp 1: g₁

comp 3|1: g₅

comp 2|1,3: g₁₁

comp 2: g₂

comp 1|2: g₆

comp 3|1,2: g₁₀

comp 2: g₂

comp 3|2: g₇

comp 1|2,3: g₁₂

comp 3: g₃

comp 1|3: g₈

comp 2|1,3: g₁₁

comp 3: g₃

comp 2|3: g₉

comp 1|2,3: g₁₂

106

Figure 2.15 Four-component series-parallel systems: (a) 2 ductile & 2 brittle Case A;

and (b) 2 ductile & 2 brittle Case B.

Ductile

Ductile

Brittle

Brittle

Ductile

Brittle

Ductile

Brittle

(a) (b)

Case A Case B

107

Figure 2.16 Effects of number of brittle components on the redundancy factor in the parallel systems consisting of (a) two components; (b) three components;

and (c) four components.

0.96

Number of brittle components in the system Two-component

Number of brittle components in the system Three-component

Number of brittle components in the system Four-component

108

Figure 2.17 Effects of post-failure behavior factor δ on redundancy factor ηR in the parallel systems consisting of (a) two components; (b) three components;

and (c) four components.

0.97

109

Figure 2.18 Effects of post-failure behavior factor δ on redundancy factor ηR in: (a) no correlation case; and (b) partial correlation case.

0.97 0.99 1.01 1.03 1.05

0 0.2 0.4 0.6 0.8 1

Redundancy factor, ηR

Post-failure behavior factor, δ Parallel system, ρ⁽R_i,R_j)=0

V(R)=0.05, V(P)=0.3 N=2 N=3

N=4 (a)

0.97 0.99 1.01 1.03 1.05

0 0.2 0.4 0.6 0.8 1

Redundancy factor, ηR

Post-failure behavior factor, δ N=3

N=4

N=2 Parallel system, ρ(R_i,R_j)=0.5

V(R)=0.05, V(P)=0.3 (b)

110

Figure 2.19 Effects of post-failure behavior factor δ on component reliability index in the parallel systems consisting of (a) two components; (b) three components; and (c) four components.

3.30

Component reliability index, βcs

Post-failure behavior factor, δ

Component reliability index, βcs

Post-failure behavior factor, δ

Component reliability index, βcs

Post-failure behavior factor, δ

111

Figure 2.20 Effects of post-failure behavior factor δ on component reliability index in the (a) no correlation case; and (b) partial correlation case.

3.30

Component reliability index, βcs

Post-failure behavior factor, δ

Component reliability index, βcs

Post-failure behavior factor, δ

112

Figure 2.21 Effects of number of components on the redundancy factor in ductile systems (Note: “N” denotes normal distribution; “LN” denotes lognormal distribution; “0” denotes ρ(Ri,Rj) = 0; “0.5” denotes ρ(Ri,Rj) = 0.5).

0.96 1.00 1.04 1.08 1.12 1.16

0 100 200 300 400 500

Redundancy factor, ηR

Number of components, N

Series systems

Parallel systems All systems

perfect correlation LN-0 N-0

N-0.5

LN-0.5

N-0 N-0.5 LN-0

LN-0.5



113

Figure 2.22 Effects of number of components on the component reliability index in ductile systems (Note: “N” denotes normal distribution; “LN” denotes lognormal distribution; “0” denotes ρ(Ri,Rj) = 0; “0.5” denotes ρ(Ri,Rj) = 0.5).

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6

0 100 200 300 400 500

Component reliability index, βcs

Number of components, N Series system N-0

Parallel system

N-0.5

LN-0 LN-0.5

All systems, perfect correlation

N-0 N-0.5 LN-0

LN-0.5



114

Figure 2.23 Failure modes of the three-component parallel system with renumbered limit state equations.

comp 1: g₁

comp 2|1: g₅

comp 3|1,2: g₉

comp 1: g₁

comp 3|1: g₆

comp 2|1,3: g₈

comp 2: g₂

comp 1|2: g₄

comp 3|1,2: g₉

comp 2: g₂

comp 3|2: g₆

comp 1|2,3: g₇

comp 3: g₃

comp 1|3: g₄

comp 2|1,3: g₈

comp 3: g₃

comp 2|3: g₅

comp 1|2,3: g₇

115

Figure 2.24 Sample space of (a) event F1; and (b) event F2.

P 2P

R₂ R₁

P 2P

0 P 2P

R₂ R₁

P 2P

0 A

B

(a) (b)

116

Figure 2.25 Effects of number of components on the redundancy factor in brittle systems (Note: “N” denotes normal distribution; “LN” denotes lognormal distribution; “0” denotes ρ(Ri,Rj) = 0; “0.5” denotes ρ(Ri,Rj) = 0.5).

0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12

0 10 20 30 40 50

Redundancy factor, ηR

Number of components, N

LN-0 (Series, parallel systems) N-0 (Series, parallel systems)

LN-0.5 (Series, parallel systems)

All systems perfect correlation N-0.5 (Series, parallel systems)

117

Figure 2.26 Effects of number of components on the reliability index of components in brittle systems (Note: “N” denotes normal distribution; “LN” denotes lognormal distribution; “0” denotes ρ(Ri,Rj) = 0; “0.5” denotes ρ(Ri,Rj) = 0.5).

3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2

0 10 20 30 40 50

Component reliability index, βcs

Number of components, N

All systems perfect correlation LN-0.5 (Series, parallel systems)

LN-0 (Series, parallel systems) N-0.5 (Series, parallel systems) N-0 (Series, parallel systems)

118

Figure 2.27 The cross-section of the bridge (dimensions are in cm).

860

20 820 20

130818

100 220 220 220 100

119

Figure 2.28 The most unfavorable longitudinal loading position of the design truck for the bridge.

4.26 m 4.26 m

35.58 kN 142.34 kN 142.34 kN 9.34 kN/m

5.74 m 5.74 m

120

Figure 2.29 The transverse position of truck wheels associated with (a) exterior girder;

and (b) interior girder for determining the lateral distribution factors (dimensions are in cm).

60 183

100 220 220

183 122

220

(a) (b)

121

Figure 2.30 Three types systems of for the analyzed bridge: (a) series system; (b) parallel system; and (c) series-parallel system.

(a)

(b)

(c) Girder 2

Girder 1 Girder 3 Girder 4

Girder 1

Girder 2

Girder 3

Girder 4

Component 1

Component 2

Girder 2 Girder 3

Girder 3 Girder 4 Girder 1

Girder 2

Component 3

Component 4

Component 5

Component 6

122

CHAPTER 3

RELIABILITY OF SYSTEMS WITH CODIFIED

In document Redundancy, Reliability Updating, and Risk-Based Maintenance Optimization of Aging Structures (Page 87-147)